Question

The sum of the rational terms in the expansion of ${(\sqrt{2}+3^{1/5})}^{10}$ is ......

Answer 1

${(2^{\frac{1}{2}}+3^{\frac{1}{5}})}^{10}$

General Term for the above series is

$T_{r+1}{=}^{10}{C}_r{\left(2^{\frac{1}{2}}\right)}^r({3^{\frac{1}{5}})}^{10-r}$

For the term to be intergral r must be the intergal mulitiple of 2 and 5

So $r=0,10$

$T_1{=}^{10}{C}_0{\left(2^{\frac{1}{2}}\right)}^0({3^{\frac{1}{5}})}^{10-0}=9T_{11}{=}^{10}{C}_{10}{\left(2^{\frac{1}{2}}\right)}^{10}({3^{\frac{1}{5}})}^{10-10}=32$

So the Sum $=32+9=41$